Integral terminal sliding mode fault tolerant control of quadcopter UAV systems

The article presents an active fault-tolerant control scheme with an integral terminal sliding mode controller for the UAV systems. This scheme effectively addresses saturation issues, disturbances, and sensor and actuator faults. Initially, the quadcopter UAV's model is represented in state space form. Subsequently, an augmented system incorporating auxiliary states from sensor faults is developed. An adaptive sliding mode observer is proposed for estimating the actuator and sensor faults. The integral terminal sliding mode fault-tolerant control, designed for altitude and attitude regulation, relies on fault estimation data. In contrast, a cascade proportional-integral-derivative (PID) controller is employed for position control. Simulation results demonstrate the superiority of the proposed method over existing control algorithms.

• The proposed method can tolerate both actuator and sensor faults in quadrotor UAV, while most of current studies can deal with only actuator faults or sensor faults.• Different to 33 , our work consider fault estimation, actuator fault, and sensor fault in the controller design as an AFTC system.Unlike 34 , this work introduces the fault estimation scheme for both sensor and actuator system.The controller uses these estimated values to provide more decisive control output for the quadrotor during the trouble.• The combination radial basis function neural network with integral sliding mode control and adaptive law not only address sensor and actuator faults but also enhance robustness against the system uncertainties.• Actuator saturation is considered in the controller design, which make the controller be more realistic and applicable to actual systems.• Stability of the closed loop system is rigorously validated using the Lyapunov theorem.Since errors in the sen- sor and actuator systems are addressed explicitly, overall system is robust to uncertainties and disturbances.
Organization of the paper is as follow.We begin by developing the quadrotor dynamic equations.Based on the attitude and altitude dynamics, the sensor and actuator fault diagnosis system are proposed in "System modeling and fault diagnosis" section.An integral sliding mode fault-tolerant controller leveraging the estimated fault signals is presented in "Fault tolerant control design" section.The effect of the control input saturation is addressed.In "Position control design" section, a simple PID control law is used for controlling the translational movement of the quadrotor.Simulations are performed in "Simulations" section under various conditions, and comparisons with existing controller are conducted."Conclusions" section concludes the study.

Modeling quadrotor dynamics
Dynamic modeling of the quadrotor has been developed in many previous works, e.g. in [8][9][10][11][12] .Essential coordinate frames of quadrotor system consist of the Earth frame (E) and the body frame (B) as shown in Fig. 1.The roll, pitch, and yaw angles are respectively defined as φ, θ, ψ ∈ (−π/2, π/2) .Also, define x, y, z ∈ R as the position coordinates of the quadrotor in E. Nonlinear dynamic model of the quadcopter can be expressed as follow: where m and control inputs U i , i = 1 to 4 are defined as is the angular speed of i th motor, g = 9.81 m/s 2 , and d φ , d θ , d ψ are disturbances in roll, pitch and yaw angle respectively.Remaining parameters of the quadrotor system are shown in Table 1.Define Then, the state equations of the attitude and altitude for the quadrotor may be expressed as (1) www.nature.com/scientificreports/where Equation (3) will be used as a basis in designing the sensor and actuator fault diagnosis system.

Fault diagnosis system design
In this section, the observer-based sensor and actuator fault diagnosis system are proposed.Since the faults occur in the attitude system due to poor actuators and noisy IMU sensor, the sensor fault diagnosis will focus solely the attitude system.From (3), state space model of the attitude and altitude system under the presence of angular rate sensor and actuator faults may be expressed as where u a and f s represent the actuator and sensor fault vector, T p is sensor fault matrix with appropriate dimension.
Later, the estimation of u a and f s will be employed by the fault tolerant controller to accommodate for the occurring fault.The following assumptions and lemma are necessary for deriving the fault diagnosis system.Assumption 1 10 The continuous nonlinear system g(x, t) is assumed to be Lipschitz, that is g(x 1 , t) − g(x 2 , t) ≤ γ �x 1 − x 2 � , where γ is the known positive constant.

Assumption 2
The external disturbance d(t) is norm bounded, i.e. d(t) < M.

Assumption 3
The pair (A p , B p ) is controllable and (A p , C p ) is detectable.
Lemma 1 21 For given a positive scalar ε and a positive definite matrix P , the following inequality holds.Lemma 2 33 For a continuous positive-definite Lyapunov function The system (4) can be reformulated as the augmented system: (3)

Let us define
Accordingly, the following adaptive fault diagnosis observer 21 is proposed to estimate the system (6)   where X is denoted as state observer vector, Ŷ is the output vector, and L 0 is the observer gain.Vector v is defined as: where δ is a small positive constant, κ is a positive gain such that κ > M .Corresponding observer error dynam- ics may be expressed as: Theorem 1 If there exist symmetric positive matrices P, G and matrices Y , K 1 , K 2 such that the following condi- tions (11-13) and adaptive law (14) hold where = PA + A T P − YC − C T Y T + I, L 0 = P −1 Y , γ and ε are positive constants.Then, the observer (8) and ( 14) can asymptotically estimate the states, sensor, and actuator faults.
Proof Consider the Lyapunov function: Using (10), the time derivative of V may be derived as From Assumption 2, we have in which f 0 and f 1 are the upper bound of ḟs and ua , and max (•), min (•) denotes the max/min eigenvalue of the matrix.Substituting ( 17)-( 21) into ( 16), we have When � < 0 , then V < −σ �ξ � 2 + β , where σ = min (−�) , which means that (e X , e u ) asymptotically con- verges to a small set around 0 according to Lyapunov stability theory.Therefore, estimation errors of the fault and the state are uniformly bounded.This proves the stability of the observer error dynamics.
It should be noted that is a standard linear matrix inequalities (LMI) form.By applying Schur complement lemma for LMI 38,39 , we can achieve the form in (11).

Fault tolerant control design
With the estimated sensor and actuator fault signals, we propose an integral terminal sliding mode fault-tolerant controller for controlling the attitude and altitude of the quadrotor.The nonlinear model of the attitude and altitude system under input saturation can be expressed as: where f s and u a are sensor and actuator fault determined from "Fault diagnosis system design" section.Define �u i = sat(u i ) − u i and �u ai = sat(u ai ) − u ai , then the system (23) can be rewritten as: V ≤ e T X (A The tracking error can be defined as: where Because ẋ1 from the sensor contains f s , time derivative of the tracking error for the sliding surface must be corrected by subtracting off with the estimate of f s , or fs , as where ẋc = x c1 x c2 x c3 x c4 T = x 2 + f s − fs .
Sliding surface for the fault-tolerant control system is defined element-wise for i = 1, . . ., 4 as: where k 1i , k 2i are positive gains; q i < p i with p i and q i are odd positive values; e is a function of time and defined as e From ( 26) and ( 27), we have: where fsi = fsi − f si .The derivative of sliding surface becomes: Typically, the uncertainty term g i is difficult to achieve in experiment.However, it can be approximated using radial basis function neural network (RBFNN) 39 as below: where W i ∈ R n is the optimal weight matrix, X i ∈ R n is the nonlinear function of hidden nodes, n = 5 , δ i is the approximation error.The Gaussian function is used for nonlinear function X i as follows: where o 2 j = 7.5 is the width of Gaussian function, r j ∈ R 2 is the center of Gaussian function which is chosen between -1 and 1, µ = e i ėi T .
Theorem 2 If the sliding surface is defined as (27) and the fault tolerant control law using sensor reading ẋ1 is designed as: and updated by: www.nature.com/scientificreports/where h 1i , h 2i , γ i are the positive gains, α i > 1, 0 < β i < 1 .Then the system (24) converges to origin in finite time.

Proof Choose the Lyapunov function as
where Wi = W i − Ŵi , Ŵi is the estimate of W i .From ( 28)-( 32), the derivative of Lyapunov function is where e ui = ûai − u ai , If we set h 1i |s i | α i ≥ η i then we have: According to (31), one obtain: Recalling Lemma 2, the terminal sliding mode surfaces ( 16)-( 21) converge to the origin in finite time.

Remark
The double reaching law in (32), ) , provides faster convergence with reduction of chattering effect 40,41 .

Position control design
In real applications, the position controller is designed with lower frequency compared to attitude and altitude controller because it is used to transform to desired roll and pitch angles.For simplicity, a cascade PID control law 42,43 is used to design the translational movements of quadrotor as Eq. ( 38) follows: where x d and y d are desired positions; x and y are current positions; k out Px , k out Py are the gains of outer loop, while Dy are the gains of inner loop.In addition, from (1) and the IMU readouts, we can determine the desired roll and pitch angles as: Figure 2 displays the overall block diagram of the control system, while Fig. 3 shows the cascade PID control law of Eq. 38.The attitude and altitude motion are controlled by the proposed fault tolerant observer and controller.The translational motion is controlled indirectly through the attitude controller by generating of the desired roll and pitch angles.

Simulations
Performance of the proposed fault diagnosis observer and fault-tolerant controller is validated through a series of numerical simulations on a quadrotor system.These simulations are crucial for demonstrating the effectiveness of the new approach under various conditions and scenarios.To provide a clear comparison and highlight the strengths of the proposed controller, the ITSMC method in 33 is used as a benchmark.The parameters of quadrotor can be summarized in Table 2.
The sampling time of simulation is chosen as T s = 0.0025s , which depends on the open-source flight control software of UAV 44 .The following parameters are chosen for fault diagnosis observer and fault-tolerant controller. (

Fault-free case
Desired translational motion is commanded as z d = 1 m , x d = 1 m , y d = 1 m at 5 , 10 , and 20 s , respectively.Desired heading is set as ψ d = 5 • at 30 s .Desired roll and pitch angles are generated through (40).Simulation is performed using our method and 33 with no fault.The tracking performance is shown in Fig. 4.Both methods show precise tracking during fault-free condition.It is noted that by adding the double reaching law, the performance of roll and pitch from proposed method is much faster than the existing method.Corresponding control inputs are plotted in Fig. 5.Note that the control inputs experience oscillation for a short interval due to step commands of the altitude and attitude angles.

Sensor fault
In this simulation, the sensor fault signal is injected into gyroscope sensor along the x-direction and z-direction as:    in Fig. 7. Estimation of the sensor fault signal is depicted in Fig. 8.The proposed fault diagnosis observer can track the actual fault signal quickly.the fault rejection can be achieved in a timely manner.The root-meansquare-error of two controllers are presented in Table 3 for comparison 45 .It is shown the proposed method is better than compared method under sensor fault.

Actuator and sensor fault
In the last simulation, actuator fault signal is introduced in addition to the earlier sensor fault signal.We assume that loss of control effectiveness (LoCE) in pitch moment and z-moment occurs at t = 60 s and t = 80 s as f apit = 0.2 and f az = 2.5 + 0.135 sin(πt/4) .The tracking performance is shown in Fig. 9.At t = 50 s where the z-moment is lost, the proposed controller still provides fast tracking performance thanks to the fault signal compensation, double reaching law.However, the method of 33 shows large deviation from the desired position due to ocssillation of roll and pitch response.Without fault signal compensation, the closed loop dynamics spends long recovery time before settling, as seen from the x-motion plot.Control inputs are plotted in Fig. 10.It should be noted that the compared method experiences large and prolonged oscillation due to multiple faults (sensor and actuator faults).Estimation of sensor and actuator faults are shown in Figs.11 and 12.The observer correctly estimate the actuator fault values and the sensor fault of sinusoidal signal generated from (40).The root-mean-square-error of both methods are presented in Table 4.It is shown the proposed method is better than compared method under sensor fault and actuator fault.

Conclusions
In this paper, an active fault tolerant control approach is proposed to resolve the effects of both actuator and sensor faults in the UAVs.We introduce the fault diagnosis observer that can estimate the sensor and actuator fault signals.Considering the control input saturation, we design an integral sliding mode fault tolerant controller that uses the estimated fault signals to compensate for the faults appropriately.Radial basis function neural network is applied in the controller to overcome the model uncertainties.The Lyapunov theorem is applied to prove the stability of the observer and controller.The efficacy of this approach is demonstrated through simulations.The results show the proposed method outperforms the baseline controller in tracking performance.This improvement is attributed to the compensation for fault effects through the fault estimation and the sliding mode fault controller with anti-saturation algorithm.Our future work will be to realize the proposed method in the actual UAV systems.

Figure 4 .
Figure 4. Tracking performance in fault-free case.

Figure 5 .
Figure 5.Control inputs in fault-free case.

Figure 6 .
Figure 6.Tracking performance in presence of sensor fault.

Figure 7 .
Figure 7.Control inputs in presence of sensor fault.

Figure 9 .
Figure 9. Tracking performance with sensor and actuator fault.

Figure 10 .Figure 11 .
Figure 10.Control inputs in presence of sensor and actuator fault.
where X

Table 4 .
Tracking errors in actuator and sensor faults.